Spectral asymptotics of periodic elliptic operators

Ola Bratteli*, Palle E.T. Jørgensen, Derek W. Robinson

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    8 Citations (Scopus)

    Abstract

    We demonstrate that the structure of complex second-order strongly elliptic operators H on Rd with coefficients invariant under translation by Zd can be analyzed through decomposition in terms of versions Hz, z ∈ Td, of H with z-periodic boundary conditions acting on L2(Id) where I = [0,1〉. If the semigroup S generated by H has a Hõlder continuous integral kernel satisfying Gaussian bounds then the semigroups Sz generated by the Hz have kernels with similar properties and z → Sz extends to a function on Cd\{0} which is analytic with respect to the trace norm. The sequence of semigroups S(m),z obtained by rescaling the coefficients of Hz by c(x) → c(mx) converges in trace norm to the semigroup Ŝz generated by the homogenization Ĥz of Hz. These convergence properties allow asymptotic analysis of the spectrum of H.

    Original languageEnglish
    Pages (from-to)621-650
    Number of pages30
    JournalMathematische Zeitschrift
    Volume232
    Issue number4
    DOIs
    Publication statusPublished - 1999

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